on tracks nature 's math

8/11/2019 On TRACKS Nature 's Math

1/20

Natures

Math

Introduction

Geometry of Spider Webs

Geometry of Spider Webs Worksheet

Measuring Trees

Measuring Trees Worksheet

Patterns in Nature

Patterns Worksheet Part A

Patterns Worksheet Part B

Vol. 18, No. 1 Kansas Wildlife & Parks Winter, 2008

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Natur e s Math

Welcome to this issue of On Tracks devoted to Natures Math. The activities included in

this issue cover spider webs, trees, and patterns in nature. These activities will address the

math standards for the state of Kansas, for the most part, within the 5th through 8th grade,with some activities even being appropriate for high school. These activities may also be

modified for younger students, especially Geometry of Spider Webs.

On T.R.A.C.K.S. 2

Math

Standard

Benchmark

Standard 1

Numbers & Computation

1 2 3 4

Standard 2Algebra

1 2 3 4

Standard 3Geometry

1 2 3 4

Standard 4Data

1 2

Geometry ofSpider Webs

x x x

Measuring

Trees

x x x x X X

Patterns in

Nature

x x x x x x x x x x


3/20

Geometry o f Spider W e b s

Objective: Students will construct their own spider web using the principles of geometry.

Background: All spiders of the same species build the same webs, but spiders from differ-ent species build different webs. Web building is instinctive for spiders. Baby spiders hatch

knowing how to design their webs. No one has to teach them how to do it.

The web of a spider can appear very complex in its design and construction,

however, observations reveal that the common garden spider constructs its web by follow-

ing a few relatively simple rules. There are three parts or components of the web:

1. radial threads, which converge in a central spot

called the hub.The radial threads are like the spokes

of a bicycle wheel

2. frame threads, which go around the outside of the

web and serve as attachment sites for the radial

threads.

3. The capture spiral.

The basic supporting structure of the orb web is provided by the radial and frame

threads, neither of which is sticky. The catching spiral, in contrast, consists of a thread stud-

ded with glue droplets.

An Arithmetic Spiral

When laying down the sticky capture spiral, the spider

starts at the outside edge, always going in one direction,

and winding inward towards the hub. At each crossing of a

radial thread, the spider fastens the sticky thread to the

radius. Both front legs and hind legs play an important role

in this process. One front leg reaches for the radius andtouches it with a quick, brushing movement to locate its

position. At the same time, one of the fourth legs pulls the

thread out of the spinnerets and dabs it against the radius.

By doing so, the spider keeps the distance between succes-

sive capture spiral turns about equal. The geometry of the

capture spiral is therefore equidistant or arithmetic.

On T.R.A.C.K.S. 3


4/20

Materials Needed:

1 skein of thread

1 thin wire clothes hangar

scissors

magic marker

a small piece of cardboard 1 x 2

Procedure:

Step 1: Take the wire coat hanger and shape the bottom triangular portion into a circle.

Step 2: With a magic marker, make a mark on the hanger at the bottom opposite the

hooked part used for hanging. Then divide each half of the circle into approximately three

equal sections. You should have six sections marked.

Step 3: Cut a 4 piece of thread. Starting at the top under the hook on the right side, attach

a thread to the wire hanger with a double knot. Moving to the left, attach the thread with a

double knot to the left side as well. These knots will only be about 1/2 apart.

Step 4: Still moving to the left, take the same thread and make a

knot at the first mark on your hanger. (This line now makes a

chord of the circle) Continue around the circle making a knot at

each mark around the circle until you reach the top starting point.

Keep your thread fairly tight as you work your way around the cir-

cle. Make a double knot at the top to secure the thread and cut off

any extra. Your hanger should be stretched with six segments of

thread like the example shown.

Step 5: Take 10-15 ft of thread from the skein and wrap it around the cardboard piece to

make it easier to work with.

Step 6: Cut a new piece of thread about 1 foot long and attach it

with a knot to the middle section of the top left thread. Stretch it

diagonally to the opposite side and knot it in the center of that

thread too. Repeat the process 2 more times, attaching a thread in

the middle of each segment and stretching it diagonally to the

opposite side until you have the center crisscrossed with threads as

shown. Trim thread ends as needed.

On T.R.A.C.K.S. 4


5/20

Step 7: Using the thread you wrapped around the cardboard

and starting in the middle of the circle (where all the threads

crisscross), secure your new thread with a double knot to keep

all the threads in the center together. When this is done, take

your thread and weave it in a circle, going from left to right,

making a left to right knot on each thread as you come to it.

(A left to right knot is like the first step in tying your shoes.

Take the string and go over and under the radial string, then

pull gently to tighten) Keep your thread fairly tight as you do

this. Keep following this same procedure as you make ever-

widening rotations around the circle. Leave about 1/2 to 3/4

between each successive row of circles that you are creating. Do this for as many rotations

as it takes to reach the outer edge of the circle. This should give you 8-12 rows, depending

on how far apart each rotation is made. If you run out of thread while you are doing this,

simply knot more string to the old and continue.

Step 8: When you get to the top (under the hook of the hanger) on your last rotation, secure

your thread with a double knot and trim any excess thread. Now you have a spider web!

What parts of a circle do

you recogn ize in your web

How a REAL spider web

i s c o ns t r uc t ed

Central

Angle

Chord

Chords are like theframe threads in an

actual spider web.

1. The first thread is asticky thread released by

the spider and carried by

the wind until it sticks to

something. Next, the spider

reinforces this bridge.

2. The spider constructs

a loose thread.

3. The spider turns the

loose thread into a Yshape forming three radii

4. The spider constructs theframe around the three radii

6. The capture spiral iswoven between the radii

Center

The center is

the hub of

the web.

This line bisects the chord at a

right angle. It acts as a radial

thread of the spider web

On T.R.A.C.K.S. 5

5. More radii are added.


6/20

Geometry o f Spider Webs

Worksheet

Name

1. Using your spider web and frame as an example, how many chords of your circle

(clothes hanger) did you create?

2. What is the polygon created by the chords?

3. Is the thread that bisects the chord equal to the diameter of the circle?

Why or Why not?

4. Draw a line in the figure below that would equal the diameter of the circle but also con-nect two opposite chords?

5. Shade half of the figure to the right to show mirror

symmetry. (There are several possibilities).

6. How could you prove that the figure from #5 has two congruent angles?

On T.R.A.C.K.S. 6


7/20

On T.R.A.C.K.S. 7

Measuring Trees

Objective: Students will measure trees in different ways and practice estimating volume called board feet.

Background: Trees, and the lumber obtained from them, are one of our most important renewable natural

resources. Foresters measure trees to help plan forest management and when to cut certain trees. By meas-

uring a trees diameter and height, one can estimate the volume of lumber for any standing tree. Todetermine the approximate amount of timber in a stand of trees, foresters do a timber cruise in which

they calculate the volume of lumber in a given area, examine the health of the forest, and count the number

of different trees found there.

Getting Ready: Making a Tree Diameter Tape. Take heavy paper or fabric and cut a strip that is 1 1/2

wide and 45 in length. (Paper strips will need to be taped together end-to-end to reach 45) Starting at

one end of your paper or fabric, mark off units 3.14 inches apart (about 3 3/16), numbering units starting

with one. This will give you a reading in inches for the trees diameter. Since you can only measure the

circumference (or distance around the tree), each 3.14 inches of circumference will equal one inch of

diameter.

1 2 3 4 5

3 3/16

6

Tree Diameter Tape

Why does measuring the circumference give me the diameter?

The circumference of a circle is the distance around the circle. It is found by the formula:

Circumference = !d d is the diameter and != 3.14

The diameter of the tree could be found by measuring the distance around the tree

with an ordinary tape measure and dividing that number by ! or 3.14

Example: If the circumference= 50, then the equation becomes:

50 = (3.14) d

50 = d

3.14

d= 15.9

By creating a tape marked off in numbers of units of ! (3.14), we shorten the process of finding the

diameter because we have already divided by !.

d


8/20

On T.R.A.C.K.S. 8

Procedure:

1. Find the diameter of the tree you have selected.

Foresters always measure the diameter of a tree at 4.5 feet (1.4 m) above the ground. This measure

ment is called Diameter at Breast Height or DBH and is the standard or rule for all foresters.

Using the tree diameter tape you just made, find the diameter of the tree 4.5 ft above the ground. This will

be the number read off the tape. Record your DBH on the worksheet.

2. Determine the height of your tree.

You will determine the height of your tree using the proportional method. Heres how:

a. Have your partner stand at the base of the tree to be measured.

b. Hold a ruler at arms length (zero end at the top) and walk backwards, keeping

your arm stiff, until the top and bottom of the ruler line up with the top and

bottom of the tree.

c. Read on the ruler where the top of your partners head appears. (For example: at 2

inches) Record on the worksheet.

d. Divide the length of the ruler by the number obtained in step c. (For example: 12inches 2 inches = 6 inches). This number becomes the proportional factor.

Record this on the worksheet.

Proportional factor = ruler length ( a )

e. Measure your partners actual height in inches. Record.

f. Multiply your partners height by the proportional factor. Record.

g. Calculate the height of the tree.

Height of student x proportional factor = Height of Tree

( d ) x ( a/b) = ( c )

EXAMPLE:

If your partners height was 55, then the height of the tree would be

55 x 6 = 330. (To find the height of the tree in feet, divide by 12)

330 12 = 27.5 feet

ruler height of person ( b )

a

b

d

c


9/20

On T.R.A.C.K.S. 9

3. Estimating board feet of lumber in your tree.

The volume of wood in a standing tree may be estimated by obtaining two measurements (diameter

and height) and applying these to a tree volume table. Volume of wood is usually measured in board feet.

A board foot is a piece of lumber that is 12 x 12 x 1 . One giant Sequoia could yield more than 500,000

board feet, enought to make 33 houses!

Use the tree volume table below to determine the number of board feet in your tree. To use the

chart, you must first divide the height of your tree by 16. The quotient of the height divided by 16 may

not be exactly a number listed on the chart so you will have to round to the nearest number to use this

chart.tree height = 27.5 ft = 1.71 (round to 1 1/2)

16 16

Example:

If your DBH is 10 and the number of 16 logs is 1 1/2, then your tree has a volume of 44 board feet.

1 / 2 1 1 1 / 2 2 2 1 / 2 3 3 1 / 2

1 0 2 1 3 4 4 4 5 5

1 2 3 0 5 2 6 8 8 5 9 8

1 4 4 2 7 4 9 9 1 2 4 1 4 3 162

1 6 5 9 1 0 0 1 3 4 1 6 9 1 9 8 2 2 6 2 4 6

1 8 7 4 1 2 9 1 7 5 2 2 1 2 5 9 2 9 7 3 2 5

2 0 9 2 1 6 2 2 2 0 2 7 9 3 2 8 3 7 7 4 1 3

DBH(inches)

Number of 16 foot logs in trees

Volume in board feet

Using Proportions to Determine Tree HeightRatios are useful ways to compare two quantities. When two ratios are equal to one another, it is called a

proportion.

There are several ways to solve a proportion. One method is to cross multiply. You can solve a propor-

tion by multiplying the numerator of the first fraction and the denominator of the second fraction and set-

ting it equal to the product of the denominator of the first fraction and the numerator of the second frac-

tion. Thus, for a proportion a = c or ad= bc

b dOur two equal ratios to determine tree height are as follows: Ruler Height of Tree = Actual Tree Height

Ruler Ht of Student Actual Student Ht

The only thing unknown in this proportion is the trees actual height. By substituting what is known in

the appropriate places, we get the following equation:

12(ruler ht of tree) = c Cross multiply and solve for c

2 (ruler ht student) 55 (actual student ht) 2c=(12)(55)

2c = 660 c= 330

When dividing the height by

16, you are estimating the

number of 16 foot pieces of

lumber you would get from

the tree.


10/20

1. Record the DBH of your tree.

Remember, this measurement is taken 4.5 feet from the base of tree)

2. While holding the ruler at arms length, what is the reading on the ruler where your

partners head is seen? This number is b, the ruler height of student.

3. Figure the proportional factor. The formula is a (ruler height of tree)

b ( ruler ht of student)

4. Record your partners actual height in inches.

5. What is the height of your tree? in inches converted to feet

Height of tree = height of student x proportional factor

c = d x a

b

6.Using your answer from #4, calculate the

number of 16 boards found in your tree.

(height of tree 16)

7. Using the tree volume table provided, find

the number of board feet of wood present

in your tree.

Measuring Trees Worksheet

Name

On T.R.A.C.K.S. 10


11/20

On T.R.A.C.K.S. 11

Patterns in NatureObjective: Students will discover the Fibonacci sequence in nature and learn of the Golden Mean alsoknown as Phi (pronounced Fi).

Background: Fibonacci was a medieval mathematician born in the year 1175

A.D. He is probably best known for a simple series of numbers introduced in hisbook,Liber abaci , and later named the Fibonacci numbers in his honor.

In Fibonaccis day, mathematical competitions and challenges were common and

in one such competition, the following problem arose:

How many pairs of rabbits will be produced in a year, beginningwith a single pair, if in every month each pair bears a new pairwhich becomes productive from the second month on?

It is easy to see that 1 pair will be produced the first month, and 1 pair also the second month (since thenew pair produced in the first month is not yet mature), and in the third month 2 pairs will be produced--

one by the original pair and one by the pair which was produced in the first month. In the fourth month, 3

pairs will be produced, and in the fifth month, 5 pairs. After this, things expand rapidly and we get the fol-

lowing sequence of

numbers:

1,1,2,3,5,8,13,21,34,55,89,144,233,377,...

Do you see the pattern?To find the next

number in the series, use the simple rule:

add the last two numbers to get the next

1,1,2,3,5,8,13,21,34,55,89,144,233,377,...

This simple, seemingly unremarkable recursive sequence has fascinated mathematicians for centuries. Its

properties show up in an amazing array of topics, from the design of Greek buildings to growth patterns in

plants.


12/20

On T.R.A.C.K.S. 12

The rabbit breeding problem that caused Fibonacci to write about the sequence inLiber abaci may

be unrealistic, but Fibonacci numbers really do appear in nature. For example, some plants branch in such

a way that they always have a Fibonacci number of growing points. Flowers often have a Fibonacci num-

ber of petals--lilies or irises have 3 petals; buttercups, wild roses and larkspur have 5 petals; delphiniums

have 8; ragwort has 13; asters and black-eyed susans have 21; and daisies may have 34, 55, or even as

many as 89 petals.Fibonacci numbers can also be seen in the arrangement of seeds on flower heads, such as that of a

sunflower. The seeds appear to be spiralling outwards both to the left and the right. Guess what? The num-

ber of spirals to the left and the number of spirals to the right are both Fibonacci numbers. Broccoli and

cauliflower, pine cones, and lettuce heads are other natural items that show Fibonacci spirals.

Fibonacci Numbers in Nature

The seeds of a sunflower form two

spirals, called parastichies, one set

eminating from the center in a clock-

wise direction, the other in a counter-

clockwise direction. In most cases, thenumber of clockwise spirals and the

number of counter clockwise spirals

will be adjacent numbers of the

Fibonacci sequence.

The reason that many seeds seem to form

spirals is that this arrangement allows the

most seeds to be packed into the space. They

are uniformly packed at any stage, all the

seeds being the same size, no crowding in

the center and not too sparse at the edges.

Left

Right

1

3

34

2

1

18

20 222419

21 23

25

26

28

29

27

30

31

33

32

17

16

15

14

13

10

9

8

567

4

11

12

15

16

17

18 19 2021

2

3

4

56

78910

11

12

13

14


13/20

On T.R.A.C.K.S. 13

Honeybees and Family Trees. There are many kinds of bees but the one best known to us is thehoneybee. It lives in a colony called a hive. There are many unusual features of the honeybee family tree

such as:

a. In a colony of honeybees, there is one special female called a queen.

She produces all the offspring in the colony.

b. There are many workerbees who are female but, unlike the queen, they do not produce

eggs.

c. There are some dronebees who are male and do no work. Males are

produced by the queens unfertilized eggs, so male bees only have a mother but no father!

d. All the females are produced when the queen has mated with a male so have two parents.

Consequently, female bees have 2parents whereas male bees have just 1parent, a female.

Lets look at the family tree of a male drone bee:

1. He had 1parent, a female.

2. He has 2 grandparents, since his mother had two parents, a male and a female.

3. He has 3 great-grandparents: his grandmother had two parents but his grandfather had only one.

Key

female

male

drone

parent

grandparents grandparents

great grandparents

great greatgrandparents

great greatgrandparents

great grandparents

Patterns in NaturePart A Fibonacci Numbers

Worksheet

Name


14/20

1. How many great-great-grandparents did the drone have? Fill in the chart below.

2. What is the pattern discovered in the family tree of honeybees?

3. We can show the Fibonacci numbers in another way. Start with two small squares of size 1 next toeach other. On top of both of these, draw a square of size 2 (1 + 1). Next, draw a new square with one side

touching the square of 2 and a square of 1--so having sides 3 units long; then another touching both a 2

square and a 3 square (so has sides of 5). Continue adding squares around the picture, each new square

having a side which is as long as the sum of the latest two squares sides. This set of rectangles who

sides are two successive Fibonacci humbers in length and which are composed of squares with sides which

are Fibonacci numbers is called the Fibonacci Rectangles. Trace the spiral drawn in the squares (a quarter

of a circle in each square.

On T.R.A.C.K.S. 14

Number of: Parents Grandparents Great

Grandparents

Great, Great

Grandparents

Great,

Great ,Great

Grandparents

MALE bees

1 2 3

FEMALE bees 2 3 5

3

8

5

2

11


15/20

On T.R.A.C.K.S. 15

While not a true mathematical spiral (like a spiders web), this is a good approximation of the kind of spi-

rals that do appear often in nature. Such spirals are seen in the shape of the shells of snails or sea shells, in

the arrangement of seeds on flowering plants (like the sunflower) and in the spirals of pinecones.

4. If the opportunity is available, gather pinecones from outside. Look at the base end where the stalk con-

nects it to the tree.Hint: Soaking the pinecones in water may make the spirals more visible.

Can you see two sets of spirals?

How many are there in each set?

Are all the cones identical in that the steep spiral (the one with the most spiral arms) goes in the same

direction?

Are the number of spirals Fibonacci numbers?

Pinecone spiralsBase endNote: Pinecone spirals below

are not of the picture above.Drawings from www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci

Spirals to the rightSpirals to the left

1

2

34

5

1 2

3

4

5

6

7

8


16/20

On T.R.A.C.K.S. 16

The Golden Mean

There is another value, closely related to the Fibonacci series, called the Golden Mean, or sometimes

the Golden Ratio. This value is obtained by making a ratio of sucessive terms in the Fibonacci series:

1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 ...1 1 2 3 5 8 13 21

If you plot a graph of these values, you will see that they seem to be tending to the same number--about

1.618. This number is denoted by the Greek letterPhi.

It seems, the Golden Mean represents an aesthetically pleasing proportion of height to width: 1 to

1.618. The ancient Greeks recognized the existence of the Golden Mean and used the ratio in the design

of one of the worlds most beautiful structures--the Parthenon.

Here is another way to look at the Golden Mean. The line below is divided according to the Golden

Mean. This is one way the Greeks used the Golden Mean in their architecture to create pleasing lines.

When we divide line AB at point C, we get the Golden Mean

The human body, as we shall see later, is also built around the Golden Mean.

..A BC

.

AB AC

AC = BC = 1.618033

Golden Mean

O = 1.618


17/20

On T.R.A.C.K.S. 17

Patterns in Nature WorksheetPart B The Golden MeanActivity adapted from Ephraim Fithiam at homepage.mac.com/efithian/Geometry. Used with permission.

The Parthenon in Greece wasbuilt using the ratio of 1: phi

Name

Examine the five rectangles drawn below from several different views. Chose the one rec-

tangle that is the most appealing to you and place an X on it. Use a metric ruler to measure

both sides of each rectangle to the nearest millimeter. For each rectangle, divide the length

of the longest side by the length of the shortest side and write this ratio on the rectangle.

1. Which rectangle was the most liked in the entire class?

2. What is its ratio?

3. Which rectangle is the Golden Mean?

4. Did you pick the Golden Mean as the most appealing?

5. What percentage of the class picked the rectangle showing the Golden Mean? (Hint: Dividethe number picked by the class by the total number in the class, then multiply this decimal number by 100 for the percentage)


18/20

Measure your height to the nearest centimeter and record it below. Measure the distance of

your navel from the floor to the nearest centimeter and record that also. Divide your height

by your navel height to find the height/navel ratio. Round this number to two decimal

places and record the ratio below.

Height = cm Navel Height= cm

6. Was your ratio 1.61?

7. What is the class average of these ratios? (Hint: Averages are found by dividing the total sum by the numberof numbers in the set)

8. Was the class average closer to the Golden Mean than your results alone?

9. Measure the lengths of the bones in your index finger to the nearest millimeter (best seen byslightly bending the finger. You will have to feel for the 4th joint.)

1st joint

2nd joint

3rd joint

4th joint

Total Length

a. Total Length 4th joint

b. 4th joint 3rd joint

c. 3rd joint 2nd joint

d. 2nd joint 1st joint

Average of a-d mm

10. Did any quotients from a-d equal 1.61?

On T.R.A.C.K.S. 18

Beautiful Body

Do Fibonacci numbers and the Golden Mean appear in the human body?

(Total Height)

(Navel Height)

Is this coincidence? Curiously,

you have 2 hands, each with 5digits, and your 8 fingers are each

comprised of 3 sections--all

Fibonacci numbers!

Ratio = cm

mm

mm

mm

mm

mm

mm

mm

mm

mm

1st Joint

2nd Joint

3rd Joint

4th Joint


19/20

On T.R.A.C.K.S. 19

Can the Golden Mean Be Heard As Well As Seen?(Activity adapted from www.enc.org/features/calendar)

The Golden Mean turned into a percent would equal 61.8 %. To find this number, we use

only the decimal portion ofPhi calledphi with a small p. Sophi is .61803 andPhi is

1.61803. To get 61.8%, we multiplyphiby 100..

It seems that from Beethoven to Mozart or Faith Hill to Rascal Flats, at 61.8% of the way

through a piece, something special almost always happens. It might be a sudden or brief

change in key, a guitar or drum solo, bridge music (a musical transition between themes), arestatement of a theme, or even silence.

Lets test the hypothesis that something often happens in a musical composition about

61.8 % of the way through it.

To Do: Provide several CDs for the class to choose selections from.

14. Figure the number of seconds 61.8% through the piece. (Hint: To do this, we multiply thetotal number of seconds the song takes to play by .618)

15. Do you agree with the statement that something happens 61.8% of the way through a

musical composition?

11. Did your average come closer to 1.61 than the individual measurements?

12. Which hand did you measure?

13. Do you think both hands would be the same?

Song ATitle:

Song BTitle:

Song CTitle:

Length of song (in seconds) Length of song (in seconds) Length of song (in seconds)

61.8% of song (in seconds) 61.8% of song (in seconds) 61.8% of song (in seconds)

Did something happen? Did something happen? Did something happen?


20/20

Editor:

Pat Silovsky

Contributing

Authors:

Pat SilovskyThis curriculum has

been modified fromone produced by the

author for USD 475

and used in after-

school programs

Editorial

Assistant:

Shelby Stevens

On TRACKS is pub-

lished by the Kansas

Department of Wildlife

& Parks several times

during the school year.

The purpose of On

TRACKS is to dissemi-

nate information and

educational resources

pertaining to the natural,

historic, and cultural

resources of the prairie,

emphasizing Kansas

ecology. Information is

presented from the per-

spective of current sci-

entific theory.

Pat Silovsky

Milford Nature Center

3115 Hatchery Dr.

Junction City, KS

(785) 238-5323

[email protected]

Alaine Neelly Hudlin

The Prairie Center

26325 W. 135th St.

Olathe, KS 66061

(913) 856-7669

[email protected]

Mike Rader

Pratt Headquarters

512 SE 25th AvePratt, KS 67124

(620) 672-0708

[email protected]

Equal opportunity to participate in and benefit from programs described herein is

available to all individuals, without regards to their race, color, national origin or

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cal affiliation. Complaints of discrimination should be sent to: Office of the

Secretary, Kansas Department of Wildlife & Parks, 1020 S. Kansas Ave, Suite 200,

Topeka, KS 66612-1327.

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